ESyMath Seminar

Introduction to tropical geometry

13:00h - Maths Common Room, UHG V3-201

Tropical geometry is a modern approach to geometry in which algebraic equations are replaced by piecewise-linear objects such as graphs and polyhedral complexes. This transformation turns geometric problems into combinatorial ones, while still preserving essential geometric information.

This talk will introduce the basic ideas of tropical geometry through examples, focusing on how tropical curves arise as simplified models of classical algebraic curves. We will also briefly discuss applications of tropical methods in areas such as enumerative geometry and interactions with other parts of mathematics.

Incidence geometry and nonlinear PDE: How to solve Schrödinger's equation by counting rectangles?

14:15h - Maths Common Room, UHG V3-201

First, I will review the classical Szemeredi-Trotter theorem, which provides an upper bound for the number of point-line incidences in the plane. Second, I will discuss Bourgain’s proof of the Strichartz estimate for the Schrödinger equation, which relies on elementary analytic number theory and Fourier analysis. Then, I will present a connection between these two topics, which leads to a sharp estimate and a global result for the cubic nonlinear Schrödinger equation.

Algebraic methods for Nash equilibrium

12:15h - A4-128/132
Joint QUAMU / ESyMath Seminar

Cumulants in the Cosmos: Mathematical Tools for Unraveling Mysteries in the Universe

10:30h - Lecture Hall H5
A joint RCM² Colloquium

This colloquium delves into two nonlinear physics problems in cosmology, each characterised by an infinite hierarchy of cumulants, essential for describing probability distributions. By examining
cosmological statistics with inherent symmetries, we leverage key mathematical theorems to simplify the path integral connecting initial and final statistics, akin to steepest-descent methods. This approach,
grounded in large deviations theory, enables us to trace the gravitational evolution of the Universe's large-scale matter structure into the nonlinear regime. Consequently, we predict significant
non-Gaussian features of the late-time density field from first principles, offering promising avenues to explore fundamental physics with forthcoming galaxy surveys. In analyzing the collisionless dynamics
of dark matter, we encounter an infinite hierarchy of coupled differential equations for the cumulants, typically truncated using fluid-like approximations. These approximations fall short in capturing
multiple fluid streams due to the exclusion of higher cumulants. We propose revisiting closure schemes based on the correspondence principle, which bridges semiclassical and classical dynamics, to
achieve an approximate closure with a finitely generated set of cumulants, rather than a finite number.

Why physicists love kagome lattices

Kagome lattices do not just make nice pictures for TRR proposals, they also constitute a paradigmatic example of frustrated quantum magnetism. In my presentation I will qualitatively explain what the fascinating magnetic properties are by contrasting them with the usual (boring) properties of bipartite antiferromagnets.

Explained with as much mathematics as I can.

12:00h - TRR Multimedia Room, UHG M4-122/126

The cost of cyclic permutations

16:15h - Lecture Hall H10
A joint RCM² Colloquium

Inplace rotation of an array involves nothing but moving data around in computer memory. As modern computer architectures involve several layers of caching, it is a surprisingly non-trivial problem. We describe a competitively fast algorithm (as indicated by measurements) and asymptotically count the number of data moves in the best, worst, and average case. It turns out that this task is equivalent to determining the expected sum of remainders encountered in a run of the Euclidean algorithm, which in turn can be estimated using tools from analytic number theory.
I shall motivate, describe and discuss this algorithm in detail an also compare it to several other reasonable choices.
(Joint work with Valentin Blomer)